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\begin{document}

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\section*{Faa di Bruno formula}

\textbf{Fa\`a{} di Bruno formula} is a remarkable combinatorial formula for higher derivatives of a composition of functions. There are various modern approaches to the related mathematics, using Joyal's theory of [[species]], operads, graphs/trees, [[combinatorial Hopf algebra]]s and so on.

\hypertarget{literature}{}\subsubsection*{{Literature}}\label{literature}

\begin{itemize}%
\item [[Imma Gálvez-Carrillo]], [[Joachim Kock]], [[Andrew Tonks]], Groupoids and Faa di Bruno formulae for Green functions in bialgebras of trees \href{http://arxiv.org/abs/1207.6404}{arxiv/1207.6404} (cf. also Kock's combinatorics \href{http://mat.uab.es/~kock/comb-pQFT.html}{page})

\end{itemize}
\begin{quote}%
We prove a Fa\`a{} di Bruno formula for the Green function in the bialgebra of P-trees, for any polynomial endofunctor P. The formula appears as relative homotopy cardinality of an equivalence of groupoids. For suitable choices of P, the result implies also formulae for Green functions in bialgebras of graphs.


\end{quote}
\begin{itemize}%
\item [[Doron Zeilberger]], \emph{Toward a combinatorial proof of the Jacobian conjecture?} in \emph{Combinatoire \'e{}num\'e{}rative} (Montreal, Que., 1985/Quebec, Que., 1985), 370--380, Lecture Notes in Math. \textbf{1234}, Springer 1986. \href{http://www.ams.org/mathscinet-getitem?mr=927775}{MR89c:05009}
\item Eliahu Levy, \emph{Why do partitions occur in Faa di Bruno's chain rule for higher derivatives?}, \href{http://arxiv.org/abs/math/0602183}{math.GM/0602183}.
\item E. Di Nardo, G. Guarino, D. Senato, \emph{A new algorithm for computing the multivariate Fa\`a{} di Bruno's formula}, \href{http://arxiv.org/abs/1012.6008}{arxiv/1012.6008}
\item Miguel A. Mendez, \emph{Combinatorial differential operators in: Fa\`a{} di Bruno formula, enumeration of ballot paths, enriched rooted trees and increasing rooted trees}, \href{https://arxiv.org/abs/1610.03602}{arXiv:1610.03602}

\end{itemize}
In works of T. J. Robinson the formula is treated in the context of vertex algebras, calculus with [[formal power series]] and in logarithmic calculus, as well as in a connection to the [[umbral calculus]]:

\begin{itemize}%
\item Thomas J. Robinson, \emph{New perspectives on exponentiated derivations, the formal Taylor theorem, and Fa\`a{} di Bruno's formula}, Proc.Conf.Vert.Op.Alg., Cont.Math. \textbf{497} (2009) 185-198 \href{http://arxiv.org/abs/0903.3391}{arxiv/0903.3391}; \emph{Formal calculus and umbral calculus}, Electronic Journal of Combinatorics, 17(1) (2010) R95 \href{http://arxiv.org/abs/0912.0961}{arxiv/0912.0961}

\end{itemize}
\hypertarget{fa_di_bruno_hopf_algebra}{}\subsubsection*{{Fa\`a{} di Bruno Hopf algebra}}\label{fa_di_bruno_hopf_algebra}

\begin{itemize}%
\item Christian Brouder, Alessandra Frabetti, Christian Krattenthaler, \emph{Non-commutative Hopf algebra of formal diffeomorphisms}, Adv. Math. \textbf{200}:2 (2006) 479-524 \href{http://www.sciencedirect.com/science/article/pii/S0001870805000265/pdf?md5=cb2367d0f3f8b07bf9799d61be34a595&pid=1-s2.0-S0001870805000265-main.pdf}{pdf}


\item Kurusch Ebrahimi-Fard, Frederic Patras, \emph{Exponential renormalization}, Annales Henri Poincare 11:943-971,2010, \href{http://arxiv.org/abs/1003.1679}{arxiv/1003.1679} \href{http://dx.doi.org/10.1007/s00023-010-0050-7}{doi}



\end{itemize}
\begin{quote}%
Using Dyson's identity for Green's functions as well as the link between the Fa\`a{} di Bruno Hopf algebra and the Hopf algebras of Feynman graphs, its relation to the composition of formal power series is analyzed.


\end{quote}
\begin{itemize}%
\item Hector Figueroa, Jose M. Gracia-Bondia, Joseph C. Varilly, \emph{Fa\`a{} di Bruno Hopf algebras}, article at Springer [[eom]], \href{http://arxiv.org/abs/math/0508337}{math.CO/0508337}


\item Jean-Paul Bultel, \emph{Combinatorial properties of the noncommutative Fa\`a{} di Bruno algebra}, J. of Algebraic Combinatorics 38:243--273 (2013) \href{http://www.ams.org/mathscinet-getitem?mr=3081645}{MR3081645}



\end{itemize}
\begin{quote}%
We give a new combinatorial interpretation of the noncommutative [[Lagrange inversion]] formula, more precisely, of the formula of Brouder--Frabetti--Krattenthaler for the antipode of the noncommutative Fa\`a{} di Bruno algebra.


\end{quote}
[[!redirects Faà di Bruno Hopf algebra]] [[!redirects Faà di Bruno formula]] [[!redirects Faà di Bruno algebra]]



\end{document}